The characteristic function of the folded normal distribution and its moment function are derived. The entropy of the folded normal distribution and the Kullback–Leibler from the normal and half normal distributions are approximated using Taylor series. The accuracy of the results are also assessed using different criteria. The maximum likelihood estimates and confidence intervals for the parameters are obtained using the asymptotic theory and bootstrap method. The coverage of the confidence intervals is also examined.

We address the problem of constraint-based causal discovery with mixed data types, such as (but not limited to) continuous, binary, multinomial, and ordinal variables. We use likelihood-ratio tests based on appropriate regression models and show how to derive symmetric conditional independence tests. Such tests can then be directly used by existing constraint-based methods with mixed data, such as the PC and FCI algorithms for learning Bayesian networks and maximal ancestral graphs, respectively. In experiments on simulated Bayesian networks, we employ the PC algorithm with different conditional independence tests for mixed data and show that the proposed approach outperforms alternatives in terms of learning accuracy.

Compositional data are met in many different fields, such as economics, archaeometry, ecology, geology and political sciences. Regression where the dependent variable is a composition is usually...

PC is a prototypical constraint-based algorithm for learning Bayesian networks, a special case of directed acyclic graphs. An existing variant of it, in the R package pcalg, was developed to make the skeleton phase order independent. In return, it has notably increased execution time. In this paper, we clarify that the PC algorithm the skeleton phase of PC is indeed order independent. The modification we propose outperforms pcalg’s variant of the PC in terms of returning correct networks of better quality as is less prone to errors and in some cases it is a lot more computationally cheaper. In addition, we show that pcalg’s variant does not return valid acyclic graphs.

The parametric Welch t-test and the non-parametric Wilcoxon–Mann–Whitney, empirical and exponential empirical likelihood tests are commonly used for hypothesis testing of two population means. In order to circumvent the inflated type I error problem of the non-parametric likelihood testing procedures, a simple calibration using the t distribution and bootstrapping is proposed. Those testing procedures are then being compared via extensive Monte Carlo simulations on the grounds of type I error and power. Evidence is provided supporting that (a) the t calibration and bootstrap improve the type I error of the non-parametric likelihoods, (b) the Welch t-test attains the type I error and produces high levels of power, and (c) the Wilcoxon–Mann–Whitney test produces inflated type I error while computation of the exact p-value is not feasible in the presence of ties. An application to real gene expression data illustrates the computational superiority of the Welch t-test.